Regular homeomorphisms of connected 3-manifolds

We establish some equivalent conditions to the Hilbert–Smith conjecture on connected n-manifolds M. For n = 3; we show that regularly almost periodic homeomorphisms of M are periodic; this result extends Theorem 5.34 of Gottschalk and Hedlund (Topological Dynamics. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1956). For the special case of ({M = mathbb{R}^3}) , we extend the result of Brechner (Pac J Math 59(2):367–374, 1975) saying that “almost periodic homeomorphisms of the plane are periodic” to ({mathbb{R}^3}) , and we show that any compact abelian group of homeomorphisms of ({mathbb{R}^3}) is either finite or topologically equivalent to a subgroup of the orthogonal group O(3).